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The Product and Quotient Rules
The Product Rule
This is another very useful formula:
d (uv) = vdu + udv
dx dx dx
This is used when differentiating a product of two functions.
Differentiate x(x² + 1)
let u = x and v = x² + 1
d (uv) = (x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 .
Again, with practise you shouldn"t have to write out u = ... and v = ... every time.
The Quotient Rule
d (u/v) = v(du/dx) - u(dv/dx)
If y = x³ , find dy/dx
x + 4
Let u = x³ and v = (x + 4). Using the quotient rule, dy/dx =
(x + 4)(3x²) - x³(1) = 2x³ + 12x²
(x + 4)² (x + 4)²